Congruent and similar shapes
We know that two shapes are similar if their corresponding angles are equal, and their corresponding sides in the same proportion. The same is true for similar triangles.
To prove that two triangles are similar, we have to show that one (not all) of the following statements is true:
1. The three sides are in the same proportion.
2. Two sides are in the same proportion, and their included angle is equal.
3. The three angles of the first triangle are equal to the three angles of the second triangle.
We have only shown two angles in the above triangles. If we calculate the missing angles, we find that angle is 55° and angle is 40°. Therefore, all corresponding angles are equal.
Are triangles PQR and SQT similar? If so, which of the above statements applies?
This question is much easier if we draw the triangles separately.
QR corresponds to QS (QR = 2QS) and QP corresponds to QT (QP = 2QT). Angle Q is the same in both triangles. Therefore, they are similar, because two sides are in the same proportion and their included angle is equal (statement 2).
Notice that they are similar, even though they are 'mirror' images (of different sizes).
Now try this one.
Are triangles ABC and DBE similar? If so, which of the above statements (1, 2 or 3) applies?
ABC and DBE are similar, because all of the corresponding angles are equal (statement 3).
If you found this difficult, remember that angle B is common to both triangles. The rules of parallel lines state that
and that (corresponding angles).
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